A characterization of bipartite Leonard pairs using the notion of a tail

Let V   denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→VA:V→V and A⁎:V→VA⁎:V→V that satisfy (i) and (ii) below.(i)There exists a basis for V with respect to which the matrix representing A   is irreducible tridiagonal and the matrix representing A⁎A⁎ is diagonal.(ii)There exists a basis for V   with respect to which the matrix representing A⁎A⁎ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Very roughly speaking, a Leonard pair is a linear algebraic abstraction of a Q-polynomial distance-regular graph. There is a well-known class of distance-regular graphs said to be bipartite and there is a related notion of a bipartite Leonard pair. Recently, M.S. Lang introduced the notion of a tail for bipartite distance-regular graphs and there is an abstract version of this tail notion. Lang characterized the bipartite Q-polynomial distance-regular graphs using tails. In this paper, we obtain a similar characterization of the bipartite Leonard pairs using tails. Whereas Lang's arguments relied on the combinatorics of a distance-regular graph, our results are purely algebraic in nature.